Introduction 2 0.1. Some theorems in mathematics with snappy model-theoretic proofs 2 1. Languages, structures, sentences and theories 2 1.1. Languages 2 1.2. Statements and Formulas 5 1.3.… Expand

Definition 1.1. For r, s ∈ R, we say that r divides s (written r|s) if there exists a t ∈ R such that s = tr. An element u ∈ R is a unit if it has a multiplicative inverse, i.e. if there exists an… Expand

We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number elds of degree 1-13. Addi- tionally we describe the algorithm used to compute these… Expand

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. The first, a close relative of a… Expand

The Alon–Füredi theorem is applied to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets.Expand

Real induction is introduced, a proof technique analogous to Mathematical Induction but applicable to statements indexed by an interval on the real line, and inductive principles in arbitrary ordered sets are pursued.Expand

We construct certain subgroups of hyperbolic triangle groups which we call \congruence" subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19… Expand

This definition could have been given to an 18th or early 19th century mathematical audience, but it would not have been very popular: probably they would not have been comfortable with the Humpty… Expand